Coastal Erosion Caused by River Mouth Migration on a Cuspate Delta: An Example from Thanh Hoa, Vietnam

Coastal erosion poses a significant threat to the infrastructure of the coastal community at the mouth of the Ma River in Thanh Hoa Province, Vietnam. In response, emergency solutions such as hard, protective structures are often implemented. However, this approach exacerbates the problem as the underlying mechanisms of coastal erosion are not adequately investigated and understood. In this study, the long-term configuration of the mouth of the Ma River in Thanh Hoa Province, Central Vietnam, is investigated using Landsat imagery spanning from 1987 to 2023. An analytical solution of a one-line model for shoreline change was also used to examine the sand discharge from the Ma River and the diffusion coefficient for the sand transported along the shore by breaking waves. The results showed an asymmetric configuration of the mouth of the Ma River over the past 37 years. The supply of sand from the Ma River is around 350,000 m3/year. The majority of sand (ranging from 55% to 75%) is mainly transported to the northern beach of the Ma River delta. This uneven distribution of sand from the Ma River has led to the asymmetrical morphology of the delta apex in which the northern part of the Ma River delta is experiencing northward movement while the southern part of the Ma River Delta is moving southward and landward. The asymmetrical morphology of the delta at the mouth of the Ma River has recently been identified as the cause of severe coastal erosion. The diffusion coefficient value determined for the transportation of longshore sand along the deltaic lobes of the Ma River delta corresponds to 90 m2/day. This study offers a practical method for investigating morphological changes in cuspate deltas, especially when measured field data are limited.

Due to urbanization and the need for natural resources, 31% of Vietnam's population lives along the coast [19]. However, this narrow coastal belt is susceptible to the threats of waves, flooding, storm surges, and sea level rise [20]. Along the coastline of Vietnam, coastal erosion has become a natural hazard in the last decade and has resulted in significant costs to both communities and nature [21][22][23][24].
The Ma River delta is a cuspate delta located in Thanh Hoa Province, Vietnam, as shown in Figure 1. The protruding part of this delta is formed by sediment which was supplied from the Ma River and is under the effects of waves from the Vietnamese East Sea [12]. According to Quang et al. [25], the Ma River delta is the third largest river delta in Vietnam, and it drains 5.17 million tons of sediment into the East Sea annually. Under the action of waves, the sediment supplied from the river is transported along the shore to nourish the beaches on both sides of the mouth of the Ma River. It should be noted that at the end of the southern coast, there is a mountainous region that acts as a boundary to block the further transport of longshore sediment (Figure 1c,d).
At the mouth of the Ma River, significant coastal erosion occurs on the cuspate of the delta. Coastal erosion has negatively affected both infrastructures and livelihoods in recent years. The cuspate delta has experienced a significant loss of agricultural land and infrastructure to the sea. Nevertheless, it is evident that numerous coastal areas within Vietnam are characterized by a restricted availability of monitoring data. Consequently, scholars specializing in coastal studies are compelled to utilize methodologies such as numerical models, physical models, or empirical formulas to conduct research, utilizing field observations gathered over a short period [26][27][28][29].
To handle the situation of limited data in the Ma River delta, an analytical solution consisting of a one-line model was utilized in combination with a remote-sensing technique to study the long-term evolution of the mouth of the Ma River in Thanh Hoa Province, Vietnam. The relationship between coastal morphological changes and coastal erosion in this cuspate delta were investigated. Based on the idea of the one-line model, this study provides a practical and efficient method of investigating morphological changes in cuspate deltas.

Application of Analytical Solutions for Shoreline Changes
The principle of a one-line model ( Figure 2) was used for the analytical approach to understanding the long-term evolution of shoreline changes in the Ma River delta. In the context of the one-line theory, it was assumed that the beach profile retains a balanced form, signifying that all underwater shapes run parallel. As a result of this assumption, a single line can be used to understand shifts in the shoreline. This chosen line conveniently corresponds to the shoreline itself, as illustrated in Figure 2. An analytical solution of a one-line model for shoreline changes in a wave-dominated delta was utilized to quickly estimate the diffusion coefficient for the longshore transport of sand (  ). Currently, there are two types of analytical solutions for modelling the wavedominated evolution of a river delta. The first solution was introduced by Larson et al. [36] for infinite shorelines (Figure 3a). Later, Duy et al. [18] derived another closed-form solution for shorelines with definite lengths (L) (Figure 3b).  [18]. In this figure, q0 is the sand supply from the river and y0 is the maximum extension of the cuspate delta into the ocean.
As can be seen in Figure 1d, there is a headland at the end of the southern shoreline in the Ma River delta. This headland can be considered a transport boundary for the further transport of sand, as shown in Figure 3b. Hence, the southern shoreline of the Ma River delta fits the analytical solution proposed by Duy et al. [18] well, as shown below: where y is the shoreline position, q0 is the sand discharged from the river,  is the diffusion coefficient, D is the sum of the depth of closure (DC) and the beach berm height (DB), x is the along-shore distance, and L is the length of the shoreline, which is the distance from the mouth of the river to the rocky boundary. In this study, we considered the rocky boundary in Figure 1d as a no-sand-transport boundary, meaning that it was assumed to prevent the movement of sand along the coastline in the longshore direction. After some time, that portion bounded by the rock will be saturated, resulting in the depletion of sediment from the system. This depletion of sediment offshore is considered a shorterm perturbation and can be ignored in the one-line model.
In order to compare the shoreline extracted from the Landsat images and the shoreline calculated using the analytical solution, a Cartesian coordinate system was used consistently in this study in which the abscissa was 108° counter-clockwise from the north and crossed the point O (595,288.00 m E; 2,188,386.00 m N) in the UTM system ( Figure  4b). In the interest of expediency, a 108° counterclockwise rotation was applied to the coordinate system to establish the configuration depicted in Figure 4a. The shoreline position extracted from the Landsat image obtained on 14 February 1991 was used as the measured data to be compared with the computed data. The shoreline position on 14 February 1991 was chosen since it has the most protruding part of the cuspate delta and a symmetric configuration with respect to the y-axis. These conditions satisfy the shoreline configuration in Figure 3b.
In order to model the shoreline position in 1991 using Equation (2), a comprehensive assembly of the requisite initial variables was undertaken, and their corresponding values are presented in Table 2. The rationale behind the selection of each variable's specific values is expounded upon in subsequent paragraphs. Quang et al. [25], reported that the Ma River discharges a total of 5.17 million tons of sediment annually into the Vietnamese East Sea. Among this amount, 20% constitutes bed load sediment [37]. Consequently, the annual contribution of bed load sediment from the Ma River to the beaches at the mouth of the river is estimated to be 5.17 × 10 6 × 20% = 1.03 × 10 6 tons. According to Quang et al. [25], the beaches of the Ma River delta are made of sand. Hence, these beaches are formed mainly via the transportation of bed load sediment from the Ma River. In determining the sand transport rate, considering a sand density of 2650 kg/m 3 [38], the calculated value is This value of q0 was utilized as the initial value for the supply of sand from the Ma River, as presented in Table 2.
The determination of the formation time (t) draws upon the research conducted by Yen et al. [39]. It is worth noting that while the investigation of Yen et al. [39] focused on the Red River delta, it also included a map encompassing the neighboring Ma River delta. This map provides distinct evidence that the historical shoreline positions in the Late Holocene originated between 500 and 200 years Before Present (BP). Consequently, the time of the establishment of the Ma River delta is considered to be approximately 500 years ago.  (4) where DC is the depth of closure, Hse and Tse are the significant wave height and wave period of the extreme wave condition (extremely high waves are expected for 12 h per year), and g is the gravitational acceleration. According to Thomson and Harris [41], Tse should be taken to be the typical period of the measured wave height. The extreme highwave conditions were taken from the study of Hung et al. [42], with Hse = 3 m and Tse = 7 s, respectively. Substituting Hse and Tse into Equation (4) yields Dc = 5.6 m. The beach berm height was calculated based on the expression proposed by Uda [43], as follows: The value of Dc = 5.6 m yields DB = 1.8 m.
Finally, the length of the shoreline was measured from the Landsat image captured on 14 February 1991 as shown in Figure 5.  Table 2 contains the initial values employed for computing the shoreline positions in 1991. A comparison between these calculated positions and the shoreline positions extracted from the Landsat image on 14 February 1991 was performed. Given that the value of ε is unknown, an iterative calculation process was adopted, considering a range of ε values. The root mean square error (RMSE) of the shoreline position was utilized to determine the optimal value of ε that best fit the data. Consequently, the iterative process terminated once the minimum RMSE was achieved.

Evolution of the Cuspate Delta
The along-shore and cross-shore coordinates of the cuspate tips on both sides of the mouth of the Ma River were observed and are used for discussing the morphological behavior of the river's mouth. The method of using the coordinates of special points such as the tips of sand spits or the center of gravity to discuss the evolution of a sand body has been applied widely in coastal engineering [27,30,32,34,44,45]. Figure 6 illustrates the schematic for measuring the coordinates of the cuspate tips, which are considered the most prominent protrusions into the open water. The schematic elucidates the fundamental variables involved in the process, namely xS and xN, signifying the along-shore coordinates of the cuspate tips on the southern and northern sectors, respectively. Additionally, the variables yS and yN were designated to represent the cross-shore extents of the cuspate tips on the southern and northern parts, respectively. It is imperative to note that all measurements of these variables are expressed in meters.

Shoreline Orientations at the River Mouth
The investigation of the mouth of the Ma River encompasses not only the geometrical characteristics of the cuspate delta but also the determination of the shoreline orientations.
In the context of wave-dominated river deltas, the assessment of the shoreline orientations on both sides of the mouth of the river assumes paramount significance for understanding the evolution of this geographical feature [2,23,46,47]. To facilitate the estimation of shoreline orientations, it is imperative to define shoreline angles. As elucidated in [46], shoreline angles correspond to the angles formed between the trendlines (red lines) and a horizontal reference line, as illustrated in Figure 7. Denoting the shoreline angles on the north and south of the mouth of the Ma River as θN and θS, respectively, the expressions for the shoreline orientations at the mouth of the river can be represented as follows: and ( )  = tan SS (7) where  N and  S are the shoreline orientations in the north and the south sections of the river mouth, respectively. In Figure 7, we examine the angles of the cuspate delta formed in the northern (θN) and southern (θS) sections. To ascertain the shoreline orientations, we employed a linear regression method, utilizing shoreline positions depicted as blue dots, which were extracted at regular 50 m intervals. For visual clarity, the figure displays shoreline positions at 300 m intervals. The measurement range for the shoreline orientations encompasses −3000 m ≤ x ≤ −1200 m for the northern shoreline and 500 m ≤ x ≤ 2500 m for the southern shoreline. According to Hu et al. [48], in wave-dominated river deltas, the shape of the coastline affects the distribution of the sand supplied from the river. This mechanism was also studied by Ashton et al. [49] and Ashton and Giosan [50]. Therefore, it would be useful to investigate the relationship between the shoreline orientations and the rate of the distribution of the sand supplied from the Ma River. If the rate of the distribution of the sand supplied from the Ma River to the north is  , it can be expressed as a function of the shoreline orientations as

Temporal Variations in Shorelines at the River Mouth
The shoreline positions extracted from the Landsat imagery are plotted in Figure 8. Only shoreline positions in 1987, 2000, 2010, and 2023 were plotted to ensure legibility. As can be seen from the figure, the southern tip of the cuspate delta retreated significantly by a distance of approximately 1000 m from 1987 to 2023. On the other hand, the shoreline position of the northern tip moved north.
Moreover, a significant northward displacement of the shoreline at the erosion hotspot (indicated by the red rectangle in Figure 8a) was also observed, measuring a distance of approximately 200 m. These shifting shoreline patterns are attributed to severe coastal erosion, which has prompted local communities to implement the use of hard structures as a protective measure, as illustrated in Figure 8b.  Figure 9 displays the optimal alignment between the computed shoreline and the observed shoreline. In this context, the shoreline position in 1991, determined through Equation (2), is depicted by a red line, while the shoreline position obtained from the Landsat image taken on 14 February 1991 is represented by blue dots (the measured shoreline). The minimum root mean square error (RMSE) between these shoreline positions amounts to 26.4 m. For reference, Table 3 presents the final values of the variables employed in the calculation procedure.   Figure 10 shows the long-term variations in the coordinates of the northern and southern sand cuspate tips over time in the along-shore direction. The horizontal axis represents the timeline, ranging from 1987 to 2023. The vertical axis represents the alongshore coordinates of the evolution of the cuspate tips, measured in meters. The blue circles on the graph represent the along-shore coordinate for the southern cuspate tip (xS), while the orange squares represent the along-shore coordinate for the northern cuspate tip (xN).

Diffusion Coefficient for Sand Transport Induced by Breaking Waves
From the data presented in Figure 10, it is evident that both cuspate tips have been moving in opposite directions along the shoreline over time. The southern cuspate tip (xS) has shown an increasing trend, moving at a rate of 17.59 m per year to the south. On the other hand, the northern cuspate tip (xN) has been decreasing in its along-shore coordinate, moving at a rate of approximately −21.28 m per year to the north. These observations are in line with the changes observed in the shoreline of the mouth of the Ma River, as depicted in Figure 8a.  Concerning the southern cuspate tip, the cross-shore coordinate of the southern cuspate tip decreased steadily over the tracking time. This means that the position of the southern cuspate tip moved closer to the shoreline as time progressed.
On the other hand, the cross-shore coordinates of the northern cuspate tip showed fluctuations during the survey period. From 1987 to 2000, the northern cuspate tip's crossshore coordinate decreased. This indicates that the position of the northern cuspate tip moved closer to the shoreline during this time period. Between 2001 and 2005, there was a sharp increase in the northern cuspate tip's cross-shore coordinate. It expanded from 2000 m to 3600 m, indicating that the northern cuspate tip moved further away from the shoreline during this period. Since 2006, the northern cuspate tip's cross-shore coordinate decreased significantly and has reached 2500 m again as of the time of writing. This suggests that the position of the northern cuspate tip moved closer to the shoreline again in recent years.
In addition, the significant increase in the northern cuspate tip's cross-shore coordinate from 2001 to 2005 indicates a large sediment supply from the Ma River, but the protrusion towards the sea of only the northern cuspate tip implies an asymmetric distribution of this sand supply, favoring the northern cuspate tip.  Figure 12 illustrates the temporal variation in the shoreline orientations at the mouth of the Ma River. The diagram displays two key values: βN, representing the shoreline orientation in the north, and βS, representing the shoreline orientation in the south.

Shoreline Orientations at the River Mouth
The diagram indicates that there is a significant gap between the shoreline orientations of the north and the south. This disparity suggests that the cuspate delta in the Ma River possesses an asymmetric shape. In other words, the two sides of the river mouth, the northern and southern tips, have different angles at which they meet the water.
Furthermore, the higher values of βN compared to the values of βS indicate that the northern tip of the river mouth protrudes more prominently into the water than the southern one. This difference in protrusion further contributes to the overall asymmetry of the cuspate delta.  Figure 13 illustrates the calculated ratio of sand supplied to the northern lobe of the Ma River delta from 1987 to 2023. The ratio was determined using Equation (8), and the results indicate that the amount of sand supplied to the northern lobe has consistently been larger than the amount of sand supplied to the southern lobe during this period. The trend over time shows a continuous increase in the sand supplied to the northern lobe, indicating a greater deposition of sand in that region compared to the south. This is evidenced by all the data points consistently lying above the midpoint (α = 0.5), suggesting an asymmetric distribution of sand towards the north (from 55% to 75%). However, there was an exception in the year 1990, when the sand supply was equal on both sides of the mouth of the Ma River, resulting in a balanced ratio with a value of α equal to 0.5. This suggests that in 1990, the distribution of sediment between the northern and southern lobes was approximately equal.

Discussion
The daily suspended sediment concentration (R) measured at the Cam Thuy Hydrological Station is plotted with the evolution of the coordinates of the southern tip in the cross-shore direction in Figure Figure 15 shows a representation of a river mouth in which sand is being supplied symmetrically to both sides of the river's opening. Additionally, the sketch includes illustrations of longshore sand transport along the coastlines. Arrows are drawn parallel to the shorelines, indicating the direction of the movement of sand along the coast.
According to the study by Duy et al. [18], longshore sand transport can be mathematically expressed as where Q is the longshore transport of sand along the coastline, and For the asymmetric condition at the Ma River delta, the sand supply from the Ma River and the ratios of the sand transported to the north and south at the mouth of the Ma River can be calculated as follows: -Sand supply from the Ma River: -Sand transported to the north: -Sand transported to the south: Figure 16 shows the sand supplied from the Ma River (orange squares) and the amounts of sand distributed to the north (blue diamonds) and the south (black dots), respectively. There is an asymmetric distribution of sand to the north and the south of the delta which indicates that the waves approach the delta from a dominant direction more often at this study site. This phenomenon can be explained by considering the wind fetch lengths, which can be observed in Figure 1a. Figure 1a distinctly reveals a pronounced disparity in fetch length between the prevailing S and SE wind directions in comparison to the NW, N, and NE sectors. As a result, it can be inferred that wave perturbations originating from the S/SE sectors are likely to induce significant longshore currents that are directed towards the NW direction and lead to the asymmetric configuration of the delta tip.
In Figure 16, it is interesting that the average amount of sand supplied by the Ma River is around 380,000 m 3 /year. This value is very close to the sand supply estimated using the analytical solution shown in Table 3 (350,000 m 3 /year).

Conclusions
Coastal erosion at the mouth of the Ma River in Thanh Hoa Province, Vietnam, was investigated using Landsat imagery from 1987 to 2023 and via analytical means. The research findings reveal the following key points: - The northern part of the Ma River delta has experienced northward movement, leading to severe coastal erosion at the mouth of the river.
-In contrast, the southern part of the Ma River Delta has been moving southward and landward. - The sand diffusion coefficient at the Ma River delta was calculated to be 90 m 2 /day, indicating the rate at which sand particles are transported. - The present investigation provides an evaluation of the supply of sand originating from the Ma River to the delta's lobes. According to the analytical solution, the estimated annual sand supply amounts to approximately 350,000 m 3 /year. Additionally, an alternative approach based on the shoreline orientations at the mouth of the river yields a slightly higher value of approximately 380,000 m 3 /year. This discrepancy underscores the reliability of the sand supply assessment derived through the methodologies utilized. - The research also highlights the asymmetric configuration of the shapes of the shoreline at the mouth of the Ma River, as seen in the uneven distribution of sediment to the delta flanks in which the ratio of sand supplied to the north ranges from 55% to 75%.
The results of this study offer significant contributions to the understanding of the coastal erosion dynamics observed at the estuary of the Ma River. These findings offer insights towards establishing prospective strategies for the coastal zone which may encompass interventions such as the implementation of jetties and harbors. Nevertheless, it is essential that antecedent to the construction of any coastal infrastructures, comprehensive research endeavors are undertaken to mitigate and attenuate any potential adverse ramifications that such constructions might impart upon the delicate coastal ecosystem [51,52].
In the absence of a cross-shore beach profile along the Ma River delta's lobes, as well as the water level measured at the mouth of the river, this study was conducted without taking into account the tidal correction for the shoreline positions. Therefore, care must be taken when using the results of this investigation.

Conflicts of Interest:
The authors declare no conflict of interest.